Cardinal utilities in the form of a vNM function allow us to resolve these problems.
Suppose an entity is faced with making a decision that can result in different outcomes \(\mathcal {O}=\lbrace o_1, \dots , o_K\rbrace \) . However, it is not deterministic which action results in which outcome. Instead, different actions can result in different lotteries over outcomes \(L \in \Delta (\mathcal {O})\) , where \(\Delta (\mathcal {O})\)  denotes the set of probability distributions over \(\mathcal {O}\) . If one wants to decide or predict which action the individual is going to make, one needs to know which action results in the most preferred lottery over the outcomes. These lotteries over outcomes are ranked by the individual's inherent preference relation \(\succeq ^{\text{vNM}}\) . [1]} showed that there exists a function \(U: \mathcal {O} \rightarrow \mathbb {R}\)  assigning a number to each outcome that is consistent with \(\succeq ^{\text{vNM}}\) , if it satisfies certain rationality axioms For a more detailed discussion of the axioms, see Chapter 5 of [2]}. The individual then chooses the action that yields the highest expected utility.
